| Møller scattering | |
|---|---|
| Name | Møller scattering |
| Caption | Feynman diagrams for lowest-order Møller scattering in Quantum electrodynamics |
| Firstreported | 1932 |
| Discoveredby | Christian Møller |
| Field | Quantum electrodynamics |
| Particles | electron–electron |
| Conserved | Electric charge, energy, momentum, lepton number, angular momentum |
Møller scattering
Møller scattering is the quantum electrodynamic process describing elastic scattering between two free electrons. It is a fundamental two-body interaction in Quantum electrodynamics (QED) that probes the structure of the electromagnetic interaction, spin-dependent forces, and radiative corrections. Møller scattering plays a central role in precision tests of Standard Model electroweak physics, beam diagnostics in accelerators such as SLAC National Accelerator Laboratory and CERN, and in determining electromagnetic form factors in low-energy experiments.
Møller scattering (e− e− → e− e−) is the prototype electron–electron scattering process, first analyzed by Christian Møller using early quantum theory and later formulated in relativistic QED. It exemplifies exchange symmetry for identical fermions and demonstrates how spin and statistics affect observable cross sections. In experimental and theoretical particle physics, Møller scattering is used for luminosity monitoring, polarimetry (measurement of electron polarization), and as a calibration channel for detectors at facilities like Jefferson Lab and DESY.
The process is sensitive to both tree-level electromagnetic interactions and higher-order electroweak corrections from exchange of virtual photons and, at higher energies, contributions involving the Z boson are non-negligible. Because electrons are identical fermions, the scattering amplitude requires antisymmetrization, leading to interference between direct and exchange amplitudes that modifies angular distributions relative to non-identical particle scattering.
In the relativistic QED framework, Møller scattering is described by perturbation theory applied to the Dirac equation coupled to the quantized electromagnetic field. The lowest nontrivial order (leading order) arises at order e^2 from two tree-level diagrams connected by photon exchange. Calculations employ covariant methods, spinor helicity states, and trace techniques using gamma matrices and the Feynman slash notation. Renormalization procedures developed in the context of Quantum field theory ensure finite predictions when including loop corrections.
The treatment of identical fermions invokes the Pauli exclusion principle and requires antisymmetrization of the total two-electron wavefunction. For polarized beams, one computes helicity-dependent matrix elements and spin density matrices, connecting theoretical predictions to observables such as analyzing powers and spin asymmetries. The formalism links directly to experimental quantities via the differential cross section dσ/dΩ in the center-of-mass or laboratory frame.
At tree level, two Feynman diagrams contribute: the t-channel (direct) photon exchange and the u-channel (exchange) photon exchange, whose amplitudes must be combined with a minus sign for fermion exchange. These diagrams are represented using the Feynman diagram formalism introduced by Richard Feynman. The invariant matrix element M can be written in terms of spinor bilinears ū(p')γ^μu(p) and the photon propagator g_{μν}/q^2.
Matrix element squared calculations employ spin sums and traces Tr[(p̸+m)γ^μ(p̸'+m)γ^ν] and exploit Mandelstam variables s, t, and u. For polarized scattering, one retains spin projection operators built from Dirac spinors or uses the helicity amplitude method pioneered in modern scattering amplitude techniques. Comparisons with analogous processes—such as Bhabha scattering (e+ e− → e+ e−)—highlight differences arising from distinct exchange channels and charge-conjugation properties.
The differential cross section for unpolarized Møller scattering in the center-of-mass frame is a function of s and scattering angle θ and shows characteristic forward–backward behavior influenced by identical-particle interference. Analytic expressions incorporate the fine-structure constant α and relativistic kinematics; at nonrelativistic energies the Rutherford-like limit is recovered with spin corrections.
Polarization profoundly affects measurable asymmetries: longitudinal and transverse beam polarizations generate spin-dependent cross sections and double-spin asymmetries used in polarimetry. Precision calculations include spinor helicity amplitudes and density matrix formalism to predict observables like the parity-conserving analyzing power A_LL. These predictions are essential for polarized-beam programs at SLAC and Thomas Jefferson National Accelerator Facility.
Møller scattering has been observed and exploited across a wide range of energies, from low-energy electron scattering experiments to high-energy collider beam diagnostics. Dedicated polarimeters use Møller scattering off magnetized iron targets to measure electron beam polarization with high precision; such devices have been implemented at CERN SPS, SLAC, and Jefferson Lab.
Precision tests of QED and electroweak theory compare measured cross sections and asymmetries with theoretical expectations including radiative and weak corrections. Møller scattering experiments have constrained physics beyond the Standard Model—e.g., limits on contact interactions and new neutral currents—complementing searches at LEP and the Large Hadron Collider. The process also serves in atomic and condensed-matter contexts for studies of low-energy electron interactions and scattering theory.
Accurate theoretical predictions require inclusion of higher-order QED corrections: one-loop virtual diagrams, real photon emission (bremsstrahlung), and vacuum polarization. Renormalized radiative corrections can be sizable and are treated using methods from perturbative renormalization theory and soft-photon resummation techniques such as the Bloch–Nordsieck approach.
Beyond QED, electroweak radiative corrections introduce Z boson exchange and box diagrams relevant at energies approaching the electroweak scale; these have been computed in the context of precision electroweak fits. At very high energies, diagrams involving heavy virtual particles predicted by extensions of the Standard Model (e.g., supersymmetry or heavy Z' bosons) modify angular distributions and asymmetries, making Møller scattering a probe for new physics. State-of-the-art calculations combine fixed-order perturbation theory with numerical Monte Carlo implementations used by experimental collaborations for event simulation and detector-level predictions.
Category:Scattering processes Category:Quantum electrodynamics